The hyperbolic-geometry tag has no wiki summary.

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### Why a non-simple geodesic in a Y-piece is NOT homotopic to a common perpendicular to the geodesic boundaries ?

This is a basic question, still I dare to ask :
Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want ...

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382 views

### Homeomorphism between the boundary of the Poincare disc S1 and its Gromov Boundary

Given the Poincare Disc $D$ and its ideal boundary $S^{1}$, I want to construct a homeomorphism between $S^{1}$ and the gromov boundary of $D$, $\partial D$, of equivalence classes of geodesics given ...

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**1**answer

244 views

### Figure eight geodesic on a pair of pants/Y-piece

Consider a figure-eight geodesic $\delta $( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $ \gamma_i$ and three perpendiculars between ...

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### Translation surfaces

I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete ...

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### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

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865 views

### Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...

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416 views

### Bolyai's construction

Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question.
It is a Compass-and-straightedge construction of asymptotically parallel line in ...

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### Length of shortest geodesic and Cheeger's isoperimetric constant for a special genus 2 surface

Let us take two copies of $ Y $-pieces [ or pair of pants ] with each boundary geodesic of length $ l $, and glue them together without any twisting to obtain a genus 2 closed orientable hyperbolic ...

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576 views

### ideal triangles in a punctured torus

I googled it and wikipidead it too : but apparently there is no definition of an ideal triangle on a punctured torus ( i.e a compact [ hyperbolic ] surface with one genus and one boundary component, ...

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697 views

### Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3?

Assume we have two geodesics in the Poincaré ball model of $\mathbb{H}^3$, viewed as arcs intersecting the boundary of and contained in the Euclidean unit sphere in $\mathbb{R}^3$. Is there a ruler ...

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### Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...

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### Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogeneously dense?

It seems to me that the results in this important paper of Kahn-Markovich imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number ...

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### Books for hyperbolic geometry ( surfaces ) with exercises ?

Hello, what are good books on hyperbolic geometry/hyperbolic surfaces that have good number of exercises, just to get a good understanding of the literature . I know John Ratcliffe's book will be one ...

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236 views

### How many pants decompositions for a given hyperbolic surface with a given hyperbolic metric

Given a hyperbolic surface of genus g ( >= 2 )and given a fixed metric on it, how many pants decompositions exist for that surface? I tend to believe that it is finite ? For example, if we take a ...

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258 views

### Questions about hyperbolic structures on a sphere with cone point singularities

How exactly do we put hyperbolic structures on a sphere with cone point singularities. Should I consider that sphere with cone points as an extended complex plane with punctures endowed with a ...

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354 views

### Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...

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873 views

### Existence of finite index torsion free subgroups of hyperbolic groups

Question. Is it true that each infinite hyperbolic group
has a torsion free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
...

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341 views

### Why do strongly irreducible Heegaard surfaces look like fibers?

I remember hearing somewhere that strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds "look like" fibers.
I know that Otal's result about short geodesics in hyperbolic mapping tori being ...

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**1**answer

696 views

### fundamental domains for free fuchsian group.

I try to understand some of the topology of the space of pointed non-compact hyperbolic surfaces (with the pointed Gromov-Hausdorff topology). It is known that the fundamental
group of a non-compact ...

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**1**answer

501 views

### Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V?

Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume $V$? Or more generally a hyperbolic $n$-manifold of volume $V$?

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### The number of cusps of higher-dimensional hyperbolic manifolds

Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.
Could ...

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**1**answer

570 views

### Finite dimensional spherical representation of $SO(n,1)(\mathbb{R})$

I'm looking for an explicit description of all the finite dimensional irreducible representation of the Lie group $SO(n,1)(\mathbb{R})$. Can you tell me, where I can find this description ? Thank you.
...

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190 views

### altering curvature on a tessellation representation of a compact surface

I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...

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280 views

### Straight line on the Poincare disk hitting points almost everywhere

Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a ...

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754 views

### Poincaré disk model: is this locus a known curve?

Please, consider a line segment $AB$ in the Poincaré disk model. Now, consider the set $S$ of all point $P$ in the disk such that the angle $\angle APB$ is constant.
Question: is $S$ a known curve?
...

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### Pythagorean Theorem for Right-Corner Hyperbolic Simplices?

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:
Euclidean: $A^2+B^2+C^2=D^2$
Hyperbolic: ...

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578 views

### Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

Hello,
Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also ...

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340 views

### Rotation part of short geodesics in hyperbolic mapping tori

Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...

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475 views

### Points at twice the distance from (-1, 0) that they are from (1, 0) in hyperbolic geometry [closed]

In answer to the question Demystifying complex numbers, Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex ...

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481 views

### The smallest Laplace-Beltrami eigenvalue on hyperbolic surfaces

For $g\geq 2$, let $M_g$ be the moduli space of genus $g$ hyperbolic surfaces, and let $\lambda_1(S_x): M_g \to \mathbb{R}$ be the smallest eigenvalue of the Laplace-Beltrami operator on the surface ...

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536 views

### Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues

Hello,
Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic ...

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646 views

### Hyperbolic structure on surfaces with boundary

I have following two questions
1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). ...

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577 views

### Reference for the geometry of horospheres

Dear all, I am looking for a reference to a proof of the following well-know fact (cited for example by
B.Farb in ``Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840).
...

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### Canonical fundamental domain for a discrete subgroup Γ of SL₂(R) acting on hyperbolic plane

Let a discrete subgroup $\Gamma$ of $SL_2(\mathbb R)$ act on the hyperbolic plane by Möbius transformations. Is there a "best" or "most canonical" fundamental domain for this action?
Some (mostly ...

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2k views

### Geodesics on a hyperbolic paraboloid

Hi,
Given any two points on a hyperbolic paraboloid ($xy = z$ or $z = (x^2 - y^2)/2$) how does one find the geodesic between them?
I know that since the hyperbolic paraboloid is doubly ruled, some ...

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### Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?

Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference ...

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### Locus of equal area hyperbolic triangles

Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...

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457 views

### Schwarz Lemma in terms of conformal surfaces or holomorphic curves?

Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...

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717 views

### What are trig classes like within a universe that's “noticeably” hyperbolic?

[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] ...

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### It is well-known that hyperbolic space is delta-hyperbolic, but what is delta?

Recall that a space is $\delta$-hyperbolic if there is some number $\delta$ with the property that every point on an edge of a geodesic triangle lies within $\delta$ of another edge. For example a ...

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2k views

### 〈x,y : x^p = y^p = (xy)^p = 1〉

Let $p$ be an odd prime and $G := \langle x,y : x^p = y^p = (xy)^p = 1 \rangle$. I want to show that $G$ is infinite and wonder if there is a good way to prove this. I'm familiar with the basics of ...

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### A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...

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### Nice proof of the triangle inequality for the metric of the hyperbolic plane

I am writing something for the journal of the university on lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \cdot ...

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### Analogy of Liouville conformal mapping theorem with Mostow rigidity?

I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar: they say that a set of geometric structures is, if ...

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### Closed hyperbolic manifold with right-angled fundamental domain

What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain?
If we allow cusps then the Whitehead link or the ...

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### Is there a volume conjecture for closed 3-manifolds?

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is
Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...

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804 views

### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...

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### cocompact discrete subgroups of SL_2

How can one construct families of cocompact discrete subgroups of the topological group $\text{SL}_2(\mathbb{C})$?
Here quaternion algebra's might help, I believe, but I have some difficulties with ...

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### Finding Constant Curvature Metrics on Surfaces without full power of Uniformization

(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)
Let S be a surface (possibly non-compact, but no boundary). It seems that there are ...

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### Weil's theorem about maps from a discrete group to a Lie group.

Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...